Moment of Inertia Calculator

Find the moment of inertia, section modulus and radius of gyration of common beam cross-sections.

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Ix (In.⁴)

Iy
Area
Section modulus Sx
Section modulus Sy
Radius of gyration rx
Radius of gyration ry
Weight
Metric Ix

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How to Calculate Moment of Inertia

The area moment of inertia (second moment of area) measures how a cross-section's material is spread away from an axis. The farther the material sits from the neutral axis, the stiffer the beam. That is why a 2×10 is far stiffer on edge than flat, and why I-beams put most of their steel in the flanges.

Rectangle: I = b × h³ ÷ 12 Circle: I = π × d⁴ ÷ 64

Hollow shapes: outside I − inside I I-beam: [bf × d³ − (bf − tw)(d − 2tf)³] ÷ 12

Section modulus S = I ÷ c Radius of gyration r = √(I ÷ A)

c is the distance from the neutral axis to the outermost fiber. Bending stress is M ÷ S, and deflection depends on E × I.

Worked Example

Tips

Lumber uses actual sizes: a 2×10 is 1.5″ × 9.25″. Rolled steel shapes have fillets and tapered flanges, so use the AISC Steel Construction Manual values for standard W, C and HSS shapes; this page is for plain geometric shapes. The x-axis here is horizontal through the center (bending under vertical loads).